I know that the velocity of a wave is given by $v=\lambda f$ but what does this velocity represent in the physical sense. For instance, if I am told a car moves at a velocity of 5 $m/s$ I know that the car itself will cover 5 meters in displacement every second. What part of the wave is moving at velocity $v$?
The velocity of a wave can be though of as the rate of change of displacement of any single peak. In more generic terms, It's the speed of a surfer who is riding the wave with no relative displacement.
The phase velocity of a wave tells us how fast any given phase of the wave is moving e.g. the crest. If we examine some fixed position, then we can count how frequently a full wavelength passes this position as the wave moves with time. This leads us directly to $v = λf$.
Note that this doesn't necessarily mean that the particles in the medium are moving at this velocity. For example, the wave could consist of particles oscillating in a transverse direction, in which case no particle has any lateral motion at all.